On multisoliton solutions of the constant astigmatism equation

TL;DR

This paper presents an algebraic method to generate infinitely many exact solutions of the constant astigmatism equation, including multisoliton solutions linked to the sine-Gordon equation, and discusses corresponding surface constructions.

Contribution

It introduces a novel algebraic formula for generating solutions of the constant astigmatism equation, extending multisoliton solutions from the sine-Gordon equation.

Findings

Infinite exact solutions derived from a seed solution

Construction of surfaces of constant astigmatism from solutions

Survey of specific multisoliton examples

Abstract

We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation $ z_{yy} + ({1}/{z})_{xx} + 2 = 0 $ from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of the constant astigmatism equation defined as counterparts of famous multisoliton solutions of the sine-Gordon equation. A few particular examples are surveyed as well.